Fig. 4: Refraction characteristics for the interface between an ordinary slab and the asymmetric phononic metamaterial.

a Color map of the refracted wave intensity as a function of the incident and refracted angles, as calculated with the full-domain 3Dd finite-element model. The black dashed line represents the refraction relation determined from the gradient of the isofrequency curves. From the colormap, negative refraction (i.e., sign(θ_i) = −sign(θ_r)) occurs for −40° < θ_i < −8° and for 0° < θ_i < 25°; moreover, the map is not invariant under the exchange of (θ_i, θ_r) with (−θ_i, −θ_r). These observations indicate the existence of asymmetric negative refraction, that happens at the angle pair identified by triangular markers. In the graph, each incident angle corresponds to a certain k_x wavevector component, as provided by the double horizontal axis. Panel b reports as colored traces the calculated isofrequency curves superimposed with the measured metamaterial reciprocal space displacement map (grayscale). The data are the same as in Fig. 3c–f, but a symmetric domain over k_y has been here employed; moreover, only the relevant sections of mode β (labeled as β*) are here highlighted. c–e Color maps illustrating the refracted beams in the various regimes enabled by the asymmetric metamaterial. The corresponding points in panels a, b are identified as green markers.